That's interesting. You are not able to delete HM moves yourself - think what would happen if you deleted Surf while surfing!
But you can take your pokemon to the Move Deleter in Mistralton City to forget a move. You can do it there because you are in a "controlled" situation and you can't get stuck anywhere.
It sounds like the same thing goes at the daycare. If an HM is deleted in the daycare there is no way to get stuck.